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G = C5×D32order 320 = 26·5

Direct product of C5 and D32

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C5×D32, C321C10, C1603C2, D161C10, C40.68D4, C20.39D8, C10.15D16, C80.19C22, C8.5(C5×D4), C4.1(C5×D8), (C5×D16)⋊5C2, C2.3(C5×D16), C16.2(C2×C10), SmallGroup(320,176)

Series: Derived Chief Lower central Upper central

C1C16 — C5×D32
C1C2C4C8C16C80C5×D16 — C5×D32
C1C2C4C8C16 — C5×D32
C1C10C20C40C80 — C5×D32

Generators and relations for C5×D32
 G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >

16C2
16C2
8C22
8C22
16C10
16C10
4D4
4D4
8C2×C10
8C2×C10
2D8
2D8
4C5×D4
4C5×D4
2C5×D8
2C5×D8

Smallest permutation representation of C5×D32
On 160 points
Generators in S160
(1 102 132 71 57)(2 103 133 72 58)(3 104 134 73 59)(4 105 135 74 60)(5 106 136 75 61)(6 107 137 76 62)(7 108 138 77 63)(8 109 139 78 64)(9 110 140 79 33)(10 111 141 80 34)(11 112 142 81 35)(12 113 143 82 36)(13 114 144 83 37)(14 115 145 84 38)(15 116 146 85 39)(16 117 147 86 40)(17 118 148 87 41)(18 119 149 88 42)(19 120 150 89 43)(20 121 151 90 44)(21 122 152 91 45)(22 123 153 92 46)(23 124 154 93 47)(24 125 155 94 48)(25 126 156 95 49)(26 127 157 96 50)(27 128 158 65 51)(28 97 159 66 52)(29 98 160 67 53)(30 99 129 68 54)(31 100 130 69 55)(32 101 131 70 56)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(97 106)(98 105)(99 104)(100 103)(101 102)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(129 134)(130 133)(131 132)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)

G:=sub<Sym(160)| (1,102,132,71,57)(2,103,133,72,58)(3,104,134,73,59)(4,105,135,74,60)(5,106,136,75,61)(6,107,137,76,62)(7,108,138,77,63)(8,109,139,78,64)(9,110,140,79,33)(10,111,141,80,34)(11,112,142,81,35)(12,113,143,82,36)(13,114,144,83,37)(14,115,145,84,38)(15,116,146,85,39)(16,117,147,86,40)(17,118,148,87,41)(18,119,149,88,42)(19,120,150,89,43)(20,121,151,90,44)(21,122,152,91,45)(22,123,153,92,46)(23,124,154,93,47)(24,125,155,94,48)(25,126,156,95,49)(26,127,157,96,50)(27,128,158,65,51)(28,97,159,66,52)(29,98,160,67,53)(30,99,129,68,54)(31,100,130,69,55)(32,101,131,70,56), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,134)(130,133)(131,132)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)>;

G:=Group( (1,102,132,71,57)(2,103,133,72,58)(3,104,134,73,59)(4,105,135,74,60)(5,106,136,75,61)(6,107,137,76,62)(7,108,138,77,63)(8,109,139,78,64)(9,110,140,79,33)(10,111,141,80,34)(11,112,142,81,35)(12,113,143,82,36)(13,114,144,83,37)(14,115,145,84,38)(15,116,146,85,39)(16,117,147,86,40)(17,118,148,87,41)(18,119,149,88,42)(19,120,150,89,43)(20,121,151,90,44)(21,122,152,91,45)(22,123,153,92,46)(23,124,154,93,47)(24,125,155,94,48)(25,126,156,95,49)(26,127,157,96,50)(27,128,158,65,51)(28,97,159,66,52)(29,98,160,67,53)(30,99,129,68,54)(31,100,130,69,55)(32,101,131,70,56), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,134)(130,133)(131,132)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148) );

G=PermutationGroup([[(1,102,132,71,57),(2,103,133,72,58),(3,104,134,73,59),(4,105,135,74,60),(5,106,136,75,61),(6,107,137,76,62),(7,108,138,77,63),(8,109,139,78,64),(9,110,140,79,33),(10,111,141,80,34),(11,112,142,81,35),(12,113,143,82,36),(13,114,144,83,37),(14,115,145,84,38),(15,116,146,85,39),(16,117,147,86,40),(17,118,148,87,41),(18,119,149,88,42),(19,120,150,89,43),(20,121,151,90,44),(21,122,152,91,45),(22,123,153,92,46),(23,124,154,93,47),(24,125,155,94,48),(25,126,156,95,49),(26,127,157,96,50),(27,128,158,65,51),(28,97,159,66,52),(29,98,160,67,53),(30,99,129,68,54),(31,100,130,69,55),(32,101,131,70,56)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(97,106),(98,105),(99,104),(100,103),(101,102),(107,128),(108,127),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118),(129,134),(130,133),(131,132),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148)]])

95 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D8A8B10A10B10C10D10E···10L16A16B16C16D20A20B20C20D32A···32H40A···40H80A···80P160A···160AF
order122245555881010101010···10161616162020202032···3240···4080···80160···160
size1116162111122111116···16222222222···22···22···22···2

95 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C5C10C10D4D8D16C5×D4D32C5×D8C5×D16C5×D32
kernelC5×D32C160C5×D16D32C32D16C40C20C10C8C5C4C2C1
# reps1124481244881632

Matrix representation of C5×D32 in GL2(𝔽31) generated by

20
02
,
027
827
,
2729
234
G:=sub<GL(2,GF(31))| [2,0,0,2],[0,8,27,27],[27,23,29,4] >;

C5×D32 in GAP, Magma, Sage, TeX

C_5\times D_{32}
% in TeX

G:=Group("C5xD32");
// GroupNames label

G:=SmallGroup(320,176);
// by ID

G=gap.SmallGroup(320,176);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,309,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D32 in TeX

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